CONSTRUCTION OF DIRECTED STRONGLY REGULAR GRAPHS USING FINITE INCIDENCE STRUCTURES

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1 CONSTRUCTION OF DIRECTED STRONGLY REGULAR GRAPHS USING FINITE INCIDENCE STRUCTURES OKTAY OLMEZ AND SUNG Y. SONG Abstract. We use finite incident structures to construct new infinite families of directed strongly regular graphs with parameters (l(q 1)q l, l(q 1)q l 1, (lq l+1)q l 2, (l 1)(q 1)q l 2, (lq l+1)q l 2 ) for integers q and l (q, l 2), and (lq 2 (q 1), lq(q 1), lq l + 1, (l 1)(q 1), lq l + 1) for all prime powers q and l {1, 2,..., q}. The new graphs given by these constructions include those with parameter sets (36, 12, 5, 2, 5), (54, 18, 7, 4, 7), (72, 24, 10, 4, 10), (96, 24, 7, 3, 7), (108, 36, 14, 8, 14) and (108, 36, 15, 6, 15) listed as feasible parameters on Parameters of directed strongly regular graphs by S. Hobart and A. E. Brouwer at http : //homepages.cwi.nl/ aeb/math/dsrg/dsrg.html. We also review these constructions and show how our methods may be used to construct other infinite families of directed strongly regular graphs. Keywords: Affine planes, group-divisible designs, partial geometries. 1. Introduction and preliminaries Directed strongly regular graphs were introduced by Duval [5] in 1988 as directed versions of strongly regular graphs. There are numerous sources for these graphs. Some of known constructions of these graphs use combinatorial block designs [9], coherent algebras [9, 14], finite geometries [8, 9, 10, 15], matrices [5, 7, 10], and regular tournaments [7, 12]. Some infinite families of these graphs also appear as Cayley graphs of groups [6, 11, 12, 13, 14]. (For the current state-of-the-art on known constructions and the existence and nonexistence results of directed strongly regular graphs are found in the Hobart and Brouwer s web [3] and the Handbook of Combinatorial Designs [4, Chap. VII.12].) In this paper, we construct some new infinite families of directed strongly regular graphs by using certain finite incidence structures, such as, nonincident point-block pairs of a divisible design and anti-flags of a partial geometry. In Section 2, we describe a construction of a directed strongly regular graph with parameters (v, k, t, λ, µ) given by (l(q 1)q l, l(q 1)q l 1, (lq l + 1)q l 2, (l 1)(q 1)q l 2, (lq l + 1)q l 2 ), for integers l 2 and q 2. The graph is defined on the set of non-incident point-block pairs of group divisible design GD(l, q l 2, q; ql). Among the feasible parameters listed in [3], our construction realizes the feasibility of the parameter sets (36, 12, 5, 2, 5) and (96, 24, 7, 3, 7); and then (72, 24, 10, 4, 10) and (108, 36, 15, 6, 15) by applying for a construction of Duval [5]. 1

2 In section 3, we construct a directed strongly regular graph on the set of anti-flags of a partial geometry. In particular, if we use the partial geometry obtained from the affine plane of order q by considering all q 2 points and taking the ql lines from exactly l pencils (parallel classes) of the plane, we obtain a directed strongly regular graph with parameters (lq 2 (q 1), lq(q 1), lq l + 1, (l 1)(q 1), lq l + 1). Thus, for example, it follows the feasibility of the parameter sets (54, 18, 7, 4, 7) and (108, 36, 14, 8, 14) among those listed in [3]. In section 4, we construct two families of directed strongly regular graphs with parameters (ql(l 1), q(l 1), q, 0, q) and (ql(l 1), 2q(l 1) 1, ql 1, ql 2, 2q) for integers q 2 and l 3. These graphs are not new. Jørgensen [13] showed that the graph with parameters (ql(l 1), q(l 1), q, 0, q) is unique for every integers l 2 and q 2. Godsil, Hobart and Martin constructed the second family of graphs in [10, Corollary 6.3]. Our construction for the second family is essentially similar to the one in [10]. However, both families of the graphs are constructed by using the almost trivial incidence structure obtained from a partition of a set of ql elements into l mutually disjoint q-element subsets. While there is a number of families of directed strongly regular graphs constructed by using flags of finite geometries (cf. [3, 9, 10, 14, 15]), there are few results for using anti-flags or partial geometries. The results presented here can only begin to illustrate some constructions using anti-flags and partial geometries or divisible designs. More work is needed to explore more general incidence structures and address the existence and non-existence problems of directed strongly regular graphs and their related incidence structures. In the remainder of this section, we recall the definition and some properties of directed strongly regular graphs. A loopless directed graph D with v vertices is called directed strongly regular graph with parameters (v, k, t, λ, µ) if and only if D satisfies the following conditions: i) Every vertex has in-degree and out-degree k. ii) Every vertex x has t out-neighbors, all of which are also in-neighbors of x. iii) The number of directed paths of length two from a vertex x to another vertex y is λ if there is an edge from x to y, and is µ if there is no edge from x to y. Another definition of a directed strongly regular graph, in terms of its adjacency matrix, is often conveniently used. Let D be a directed graph with v vertices. Let A denote the adjacency matrix of D, and let I = I v and J = J v denote the v v identity matrix and all-ones matrix, respectively. Then D is a directed strongly regular graph with parameters (v, k, t, λ, µ) if and only if (i) JA = AJ = kj and (ii) A 2 = ti + λa + µ(j I A). 2

3 Duval observed that if t = µ and A satisfies above equations (i) and (ii), then so does A J m for every positive integer m; and so, we have: Proposition 1.1. [5] If there exists a directed strongly regular graph with parameters (v, k, t, λ, µ) and t = µ, then for each positive integer m there exists a directed strongly regular graph with parameters (mv, mk, mt, mλ, mµ). Proposition 1.2. [5] A directed strongly regular graph with parameters (v, k, t, λ, µ) has three distinct integer eigenvalues with multiplicities θ 0 = k, θ 1 = 1 2 (λ µ + δ), θ 2 = 1 (λ µ δ) 2 m 0 = 1, m 1 = k + θ 2(v 1) θ 1 θ 2, m 2 = k + θ 1(v 1) θ 1 θ 2, respectively, where δ = (µ λ) 2 + 4(t µ) is a positive integer. Throughout the paper, we will write x y if there is an edge from a vertex x to another vertex y, and x y if there is no edge from x to y. We will also write x y if and only if both x y and y x. 2. Construction of graphs using certain divisible designs The first family of directed strongly regular graphs, which we shall describe in this section, use non-incident point-block pairs of group divisible designs GD(l, q l 2, q; ql) for integers q 2 and l 2. Let P be a ql-element set with a partition P of P into l parts ( groups ) of size q. Let P = {S 1, S 2,..., S l }. Let B = {B P : B S i = 1 for all i = 1, 2,..., l}. Then B consists of q l subsets of size l called blocks. The elements of P will be called points of the incidence structure (P, B) with the natural pointblock incidence relation. This structure has property that any two points from the same group never occur together in a block while any two points from different groups occur together in q l 2 blocks. It is known as a groupdivisible design GD(l, q l 2, q; ql). Definition 2.1. Let (P, B) be the incidence structure defined as above. Let D = D(P, B) be the directed graph with its vertex set V (D) = {(p, B) P B : p / B}, and directed edges given by (p, B) (p, B ) if and only if p B. Theorem 2.1. Let D be the graph D(P, B) defined in Definition 2.1. Then D is a directed strongly regular graph with parameters v = lq l (q 1), k = lq l 1 (q 1), t = µ = q l 2 (lq l + 1), λ = q l 2 (l 1)(q 1). 3

4 Proof: It is easy to verify the values of v and k. To find the value of t, let (p, B) V (D) with B = {b 1, b 2,..., b l }, and let N + ((p, B)) and N ((p, B)) denote the set of out-neighbors and that of in-neighbors of (p, B), respectively. Then for t we need to count the elements of N + ((p, B)) N ((p, B)) = {(p, B ) V (D) : p B, p B}. Without loss of generality, suppose that p S 1. Then condition p B requires that b 1 must be p. The second condition p B requires that p = b i for some i. With the choice of p = b 1, there are q l 1 blocks B = {b 1, b 2,..., b l } with b 1 = p can be paired with p = b 1. On the other hand, with p = b j for j 1, we have (q 1)q l 2 choices for B with b 1 = p, b j S j \ {b j } and b i S i for all i {2, 3,..., l} \ {j}. Hence t = q l 1 + (l 1)(q 1)q l 2. We now claim that the number of directed paths of length two from a vertex (p, B) to another vertex (p, B ) depends only on whether there is an edge from (p, B) to (p, B ) or not. Suppose that (p, B) (p, B ); that is, p B. Then, without loss of generality, we may assume that p = b 1, and have λ = N + ((p, B)) N ((p, B )) = {(p, B ) V (D) : p = b 1 B, p B } = {(p, {p, b 2,..., b l }) : p B \ {b 1 }, b i S i \ {p } for i = 2, 3,..., l} = (l 1)(q 1)q l 2. For µ, suppose (p, B) (p, B ). Then µ = {(p, B ) : p B, p B } with p / B. Without loss of generality, we assume that p S 1, and have µ = {(b 1, {p, b 2,..., b l }) : b i S i for i = 2, 3,..., l} + l {(b i, {p, b 2,..., b l }) : b i S i \ {b i }, b j S j for j {2, 3,..., l} \ {i}} i=2 = q l 1 + (l 1)(q 1)q l 2. This completes the proof. Remark 2.1. By Proposition 1.1, there are directed strongly regular graphs with parameters (mlq l (q 1), mlq l 1 (q 1), mq l 2 (lq l+1), mq l 2 (l 1)(q 1), mq l 2 (lq l+1)) for all positive integers m. These graphs can be constructed directly by replacing all edges by multiple edges with multiplicity m in the above construction. Remark 2.2. In the case of l = 2 (m = 1) in the above construction, we obtain the graphs with parameters (2q 2 (q 1), 2q(q 1), 2q 1, q 1, 2q 1). These graphs are constructed on the sets of non-incident vertex-edge pairs of complete bipartite graphs K q,q. If we use complete bipartite multigraph with multiplicity m for each edge as in the above remark, we obtain directed strongly regular graphs with parameters, (2mq 2 (q 1), 2mq(q 1), m(2q 1), m(q 1), m(2q 1)). 4

5 With several combinations of small m and q, we obtain new directed strongly regular graphs with parameter sets (36, 12, 5, 2, 5), (96, 24, 7, 3, 7), (72, 24, 10, 4, 10) and (108, 36, 15, 6, 15). So the feasibility of these parameter sets which are listed in [3] has been realized. By Proposition 1.2, we can easily compute the eigenvalues for D(P, B). Lemma 2.2. If a directed strongly regular graph with parameters (v, k, t, λ, µ) satisfies t = µ = λ + q l 1 for positive integers q and l 2, then its eigenvalues are θ 0 = k, θ 1 = 0, θ 2 = q l 1 with multiplicities m 0 = 1, m 1 = v 1 k q l 1, m 2 = k q l 1, respectively. Example 2.3. (Parameter sets for small DSRGs (orders up to 110) constructed.) (8, 4, 3, 1, 3) l = q = 2 (16, 8, 6, 2, 6) l = q = 2; m = 2 (24, 12, 9, 3, 9) l = q = 2; m = 3 (24, 12, 8, 4, 8) l = 3, q = 2 (32, 16, 12, 4, 12) l = q = 2; m = 4 (36, 12, 5, 2, 5) l = 2, q = 3 (40, 20, 15, 5, 15) l = q = 2; m = 5 (48, 24, 16, 8, 16) l = 3, q = 2; m = 2 (48, 24, 18, 6, 18) l = q = 2; m = 6 (56, 28, 21, 7, 21) l = q = 2; m = 7 (64, 32, 20, 4, 20) l = 4, q = 2 (64, 32, 24, 8, 24) l = q = 2; m = 8 (72, 24, 10, 4, 10) l = 2, q = 3; m = 2 (72, 36, 24, 12, 24) l = 3, q = 2; m = 2 (72, 36, 27, 9, 27) l = q = 2; m = 9 (80, 40, 30, 10, 30) l = q = 2; m = 10 (88, 44, 33, 11, 33) l = q = 2; m = 11 (96, 24, 7, 3, 7) l = 2, q = 4 (96, 48, 32, 16, 32) l = 3, q = 2; m = 4 (96, 48, 36, 12, 36) l = q = 2; m = 12 (104, 52, 39, 13, 39) l = q = 2; m = 13 (108, 36, 15, 6, 15) l = 2, q = 3; m = 3 3. Construction of graphs using partial geometries In this section, we use partial geometries to construct a family of directed strongly regular graphs. The concept of a partial geometry was introduced by R. C. Bose in connection with his study of large cliques of more general strongly regular graphs in [1]. A partial geometry pg(κ, ρ, τ) is a set of points P, a set of lines L, and an incidence relation between P and L with the following properties: (i) Every line is incident with κ points (κ 2), and every point is incident with ρ lines (ρ 2). (ii) Any two points are incident with at most one line. (iii) If a point p and a line L are not incident, then there exists exactly τ (τ 1) lines that are incident with p and incident with L. 5

6 Here we use parameters (κ, ρ, τ) instead of more traditional notations, (K, R, T ) or (1 + t, 1 + s, α) used in [1, 2] or [16]. In what follows, we often identify a line L as the set of κ points that are incident with L; so, write p L, as well as p is on L, and L passes through p when p and L are incident. Definition 3.1. Let D = D(pg(κ, ρ, τ)) be the directed graph with its vertex set V (D) = {(p, L) P L : p / L}, and directed edges given by (p, L) (p, L ) if and only if p L. Theorem 3.1. Let D be the graph D(pg(κ, ρ, τ)) defined as above. Then D is a directed strongly regular graph with parameters ( ) v = κρ(κ 1)(ρ 1) τ 1 + (κ 1)(ρ 1) τ, k = κρ(κ 1)(ρ 1) τ, t = µ = κρ τ, λ = (κ 1)(ρ 1). Proof: The value ( v is clear as ) it counts the number ( of anti-flags ) of pg(κ, ρ, τ) which has κ 1 + (κ 1)(ρ 1) τ points and ρ 1 + (κ 1)(ρ 1) τ lines. Also it is clear that k = {(p, L ) : p L } = ρ(v κ) = κρ(κ 1)(ρ 1)/τ. For t, given a vertex (p, L) V (D), we count the cardinality of N + ((p, L)) N ((p, L)) = {(p, L ) V (D) : p L, p L}. Among the ρ lines passing through p, ρ τ lines are parallel to L. If L is parallel to L, then all κ points on L can make legitimate non-incident point-line pairs (p, L ) with given L. In the case when L is not parallel to L, all points on L except for the common incident point of L and L, can form desired pairs. Hence we have t = (ρ τ)κ + τ(κ 1) = κρ τ. For λ, suppose (p, L) (p, L ). It is clear that λ = {(p, L ) : p L, p L } = (ρ 1)(κ 1), because each of ρ 1 lines passing through p (excluding L ) can be paired with any of κ 1 points on L \ {p}. For given (p, L) and (p, L ) with p / L, µ = {(p, L ) : p L, p L } = (ρ τ)κ + τ(κ 1) = κρ τ, since among the ρ lines passing through p, the ones that are parallel to L can form desired pairs with any of κ points on L, while each of the remaining τ lines can be paired with κ 1 points on L. Partial geometries are ubiquitous. For example, partial geometries with τ = 1 are generalized quadrangles, those with τ = κ 1 are transversal designs, and those with τ = ρ 1 are known as nets. In order to have some concrete examples of new directed strongly regular graphs of small order, we 6

7 consider a special class of partial geometries that are obtained from finite affine planes. Let AP (q) denote the affine plane of order q. Let AP l (q) denote the partial geometry obtained from AP (q) by considering all q 2 points and taking the lines of l parallel classes of the plane. Then AP l (q) inherits the following properties from AP (q): (i) every point is incident with l lines, and every line is incident with q points, (ii) any two points are incident with at most one line, (iii) if p and L are non-incident point-line pair, there are exactly l 1 lines containing p which meet L. That is, AP l (q) = pg(q, l, l 1). Corollary 3.2. Let D = D(AP l (q)) be the directed graph D(pg(q, l, l 1)) defined as in Definition 3.1. Then D is a directed strongly regular graph with parameters (v, k, t, λ, µ) = (lq 2 (q 1), lq(q 1), lq l + 1, (l 1)(q 1), lq l + 1). Proof: It immediately follows from Theorem 3.1. In particular, if l = q, AP q (q) = pg(q, q, q 1) is a transversal design TD(q, q) = (P, G, L) of order q, block size q, and index 1 in the following sense. (i) P is the set of q 2 points of AP q (q); (ii) G is the partition of P into q classes (groups) such that each class consists of q points that were collinear in AP (q) but not in AP q (q); (iii) L is the set of q 2 lines (blocks); (iv) every unordered pair of points in P is contained in either exactly one group or in exactly one block, but not both. Corollary 3.3. Let D be the graph D(AP q (q)) defined as above. Then D is a directed strongly regular graph with parameters (v, k, t, λ, µ) = (q 3 (q 1), q 2 (q 1), q 2 q + 1, (q 1) 2, q 2 q + 1). The eigenvalues of this graph are q 2 (q 1), 0, q with multiplicities 1, q 4 q 3 q 2 + q 1, q(q 1), respectively. Example 3.1. This method produces the DSRGs with the following parameter sets (with order up to 110 and m = 1 only): (8, 4, 3, 1, 3) l = q = 2 (12, 6, 4, 2, 4) l = 3, q = 2 (16, 8, 5, 3, 5) l = 4, q = 2 (20, 10, 7, 4, 7) l = 5, q = 2 (24, 12, 7, 5, 7) l = 6, q = 2 (28, 14, 8, 6, 8) l = 7, q = 2 (32, 16, 9, 7, 9) l = 8, q = 2 (36, 12, 5, 2, 5) l = 2, q = 3 (54, 18, 7, 4, 7) l = q = 3 (72, 24, 9, 6, 9) l = 4, q = 3 (96, 24, 7, 3, 7) l = 2, q = 4 7

8 Among these, the parameter sets (54, 18, 7, 4, 7); and so (108, 36, 14, 8, 14), (162, 54, 21, 12, 21) confirm the feasibility of the putative parameter sets listed in [3]. These graphs share the same automorphism group which is isomorphic to (((Z 3 Z 3 ) Z 3 ) Z 2 ) Z 2. Remark 3.2. When l = 2, we obtain the directed strongly regular graph with parameters (4q 2, 4q, 2q 1, q 1, 2q 1) in both Section 2 and Section 3. The two graphs are shown to be isomorphic although the construction methods are different. We demonstrate the isomorphism through an example; namely, for the case when q = 3, the directed strongly regular graph with parameters (36, 12, 5, 2, 5). The one, denoted by D 1, is coming from non-incident vertexedge pairs of the complete bipartite graph K 3,3 with partite sets {1, 2, 3} and {4, 5, 6}. The other, denoted by D 2, comes as D 2 = D(AP ) 2 (3), with point set P = {1, 2,..., 9} and line set L = {123, 456, 789, 147, 258, 369} of AP 2 (3). The following map between the vertices (anti-flags of the underlying incident structures,) establishes the isomorphism between the graphs D 1 and D 2, where the adjacency of vertices in D 1 is defined by while that of D 2 is defined by (h, ij) (h, i j ) iff h i j (hij, l) (h i j, l ) iff l {hij}. 1, , 4 2, , 1 3, , 1 1, , 5 2, , 2 3, , 2 1, , 6 2, , 3 3, , 3 1, , 7 2, , 7 3, , 4 1, , 8 2, , 8 3, , 5 1, , 9 2, , 9 3, , 6 4, , 2 5, , 1 6, , 1 4, , 3 5, , 3 6, , 2 4, , 5 5, , 4 6, , 4 4, , 6 5, , 6 6, , 5 4, , 8 5, , 7 6, , 7 4, , 9 5, , 9 6, , 8 4. Construction of graphs using partitioned sets In this section we construct directed strongly regular graphs for two parameter sets, (ql(l 1), q(l 1), q, 0, q) and (ql(l 1), 2q(l 1) 1, ql 1, ql 2, 2q) for all positive integers q 1 and l 3. The incidence structure which will be used here may be viewed as a degenerate case of those used in earlier sections. The graphs produced by this construction are not new. The latter family of graphs have been constructed by Godsil, Hobart and Martin in [10, Corollary 6.3]. Nevertheless, we introduce our construction to illustrate 8

9 a variation of constructions which may be applied to produce new families of directed strongly regular graphs. Let P be a set of ql elements ( points ), and let S 1, S 2,..., S l be l mutually disjoint q-element subsets of P ( blocks ). We denote the family of blocks by S = {S 1, S 2,..., S l }. We will say that point x P and block S S is a non-incident point-block pair if and only if x / S. For the directed strongly regular graph with the first parameter set, let D = D(P, S) be the directed graph with its vertex set V (D) = {(x, S) P S : x / S}, and directed edges defined by (x, S) (x, S ) if and only if x S. Theorem 4.1. Let P, S and D(P, S) be as the above. Then D(P, S) is a directed strongly regular graph with parameters (ql(l 1), q(l 1), q, 0, q). Proof: Easy counting arguments give the values for parameters. Remark 4.1. Jørgensen [13] showed that the directed strongly regular graph with parameters (ql(l 1), q(l 1), q, 0, q) is unique for all positive integers l and q. For the directed strongly regular graph with the second parameter set, let G = G(P, S) be the directed graph with its vertex set V (G) = {(x, S) P S : x / S}, where P and S are as the above, and let edges on V (G) be defined by: (x, S) (x, S ) if and only if { (1) x S ; or (2) S = S and x x. Theorem 4.2. Let G be the graph G(P, S) defined as above. Then G is a directed strongly regular graph with parameters (ql(l 1), 2q(l 1) 1, ql 1, ql 2, 2q). Proof: Clearly v = ql(l 1) as V (G) = V ( S 1). Given a vertex (x, S), let N 1 ((x, S)) := {(x, S ) : x S and x S} N 2 ((x, S)) := {(x, S ) : S = S and x x } N 3 ((x, S)) := {(x, S ) : x S and x / S} Then by simple counting, we have N 1 ((x, S)) = q N 2 ((x, S)) = q(l 2) + q 1 N 3 ((x, S) = q(l 2). Hence, t = N 1 ((x, S)) + N 2 ((x, S)) = ql 1, and k = t + N 3 ((x, S)) = 2ql 2q 1. Finally, we count the number of vertices (x, S ) that belongs to N + ((x, S)) N ((x, S )) for λ and µ. 9

10 For λ, given an edge (x, S) (x, S ), we shall need to consider the following two cases separately: (Case 1) when x S, and (Case 2) when S = S and x x. Case (1). Suppose x S. Then unless S = S, S must contain x which forces S = S. With S = S, all points that are not belong to S {x } can be chosen to be x ; and so, there are q(l 1) 1 choices for x. With S = S, the q 1 points of the set S \ {x} are possible for x. Together there are ql 2 vertices (x, S ) belonging to N + ((x, S)) N ((x, S )). Case (2). Let S = S. Then with the choice of S = S = S, x can be chosen from P \ (S {x, x }); and so, there are q(l 1) 2 choices for x. Also with the choice of S being the block containing x, any point in S can be chosen as x ; and so, there are q possible choices for x. Together we have ql 2 choices for (x, S ) as well. Hence we have λ = ql 2. For µ, suppose S S and x / S. Then it is clear that µ = {(x, S ) : x S, x S } {(x, S ) : S = S, x S } = q + q. This completes the proof. References [1] R. C. Bose. Strongly regular graphs, partial geometries, and partially balanced designs, Pacific J. Math. 13(1963) [2] A. E. Brouwer, A. M. Cohen and A. Neumaier. Distance-Regular Graphs, Springer- Verlag, Berlin, [3] A. E. Brouwer and S. Hobart. Parameters of directed strongly regular graphs. /ãeb/math/dsrg/dsrg.html [4] A. E. Brouwer and S. Hobart. Directed strongly regular graph, in: C. Colburn and J. Dinitz (Eds.), Handbook of Combinatorial Designs, CRC Inc., Boca Raton, [5] A. Duval. A Directed Graph Version of Strongly Regular Graphs. Journal of Combinatorial Theory, Series A 47 (1988), [6] A. Duval and D. Iourinski. Semidirect product constructions of directed strongly regular graphs, Journal of Combinatorial Theory, (A) 104 (2003) [7] C. Eschenbach, F. Hall, R. Hemasinha, S. Kirkland, Z. Li, B. Shader, J. Stuart, J. Weaver. Properties of Tournaments among Well-Matched Players. The American Mathematical Monthly, 107 (2000), [8] F. Fiedler, M. Klin and M. Muzychuk. Small vertex-transitive directed strongly regular graphs, Discrete Mathematics 255 (2002) [9] F. Fiedler, M. Klin, Ch. Pech. Directed strongly regular graphs as elements of coherent algebras, in: K. Denecke, H.-J. Vogel (Eds.), General Algebra and Discrete Mathematics, Shaker Verlag, Aachen, 1999, pp [10] C. D. Godsil, S. A. Hobart and W. J. Martin. Representations of directed strongly regular graphs. European J. Combin. 28 (2007), no. 7, [11] S. Hobart, and T. Shaw. A Note on a Family of Directed Strongly Regular Graphs. European Journal of Combinatorics 20 (1999), [12] L. Jørgensen. Directed Strongly Regular Graphs with µ = λ. Discrete Mathematics, 231 (2001), no. 1 3, [13] L. Jørgensen. Non-existence of directed Strongly Regular Graphs. Discrete Mathematics, 264 (2003), [14] M. Klin, A. Munemasa, M. Muzychuk and P.-H. Zieschang. Directed strongly regular graphs obtained from coherent algebras, Linear Algebra Appl. 377 (2004), [15] M. Klin, C. Pech and P.-H. Zieschang. Flag algebras of block designs: I. Initial notions, Steiner 2-designs, and generalized quadrangles, preprint MATH-AL , Technische Universitat Dresden. 10

11 [16] J. A. Thas. Partial geometries, in: C. Colburn and J. Dinitz (Eds.), Handbook of Combinatorial Designs, CRC Inc., Boca Raton, Department of Mathematics, Iowa State University, Ames, Iowa, 50011, U. S. A. address, O. Olmez: Department of Mathematics, Iowa State University, Ames, Iowa, 50011, U. S. A. address, S. Y. Song: 11